NotAwfulTech

1. Don't worry about the sequences changing direction. Just call the check function both assuming it is increasing and assuming it is decreasing. This is cheap enough because the wrong branch will fail after 3 elements or so.
2. When encountering an element that fails, you only need to consider removing the previous element, or the current element. If you can get to the next element removing one of those then you can continue on without any real backtracking.

#include <string>
#include <iostream>
#include <sstream>
#include <vector>
#include <iterator>

bool valid_pair(const std::vector<int> &arr, int i, int j, bool direction) {
  if (i < 0) return true;
  if (j == arr.size()) return true;
  return    !(arr[i] == arr[j])
         && (direction ? arr[i] < arr[j] : arr[j] < arr[i])
         && (std::abs(arr[j]-arr[i]) <= 3);
}

bool valid(const std::vector<int> &arr, bool direction) {
  int checks = 1;
  for (int i = 1; i < arr.size(); ++i) {
    if (valid_pair(arr, i-1, i, direction)) continue;
    if (checks == 0) return false;
    if (   valid_pair(arr, i-2,  i, direction)
        && valid_pair(arr, i,  i+1, direction)) {
      checks -= 1; i += 1;
    } else if (valid_pair(arr, i-1, i+1, direction)) {
      checks -= 1; i += 1;
    } else return false;
  }
  return true;
}

int main() {
  int safe = 0;
  std::string s;
  while (std::getline(std::cin, s)) {
    std::istringstream iss(s);
    std::vector<int> report((std::istream_iterator<int>(iss)),
                            std::istream_iterator<int>());
    safe += (valid(report, true) || valid(report, false));
  }
  std::cout << safe << std::endl;
}